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We present a spectroscopic survey of Ly$\alpha$ emitters (LAEs) at $z\sim3.7$and $z\sim4.8$. The LAEs are selected using the narrowband technique based onthe combination of deep narrowband and broadband imaging data in two deepfields, and then spectroscopically confirmed with the MMT multi-fiberspectrograph Hectospec. The sample consists of 71 LAEs at $z\sim3.7$ and 69LAEs at $z\sim4.8$ over $\sim 1.5$ deg$^2$, making it one of the largestspectroscopically confirmed sample of LAEs at the two redshifts. TheirLy$\alpha$ luminosities are measured using the secure redshifts and deepphotometric data, and span a range of $\sim 10^{42.5}$ - $10^{43.6} \,\rm erg\,s^{-1}$, so these LAEs represent the most luminous galaxies at the redshifts interms of Ly$\alpha$ luminosity. We estimate and correct sample incompletenessesand derive reliable Ly$\alpha$ luminosity function (LF)s at $z\sim3.7$ and 4.8based on the two spectroscopic samples. We find that our Ly$\alpha$ LFs areroughly consistent (within a factor of $2-3$) with previous measurements atsimilar redshifts that were derived from either photometric samples orspectroscopic samples. By comparing with previous studies in differentredshifts, we find that the Ly$\alpha$ LFs decrease mildly from $z\sim3.1$ to$z\sim5.7$, supporting the previous claim of the slow LF evolution between$z\sim2$ and $z\sim6$. At $z>5.7$, the LF declines rapidly towards higherredshift, partly due to the effect of cosmic reionization.

A spectroscopic survey of Ly$\alpha$ emitters and Ly$\alpha$ luminosity function at Redshifts 3.7 and 4.8